Question

Prove the following number is irrational/not rational: $5+3\sqrt{2}$

Answer 1

Ir-rational number:

To prove that $5+3\sqrt{2}$ is an irrational number we will use contradiction method.

Let us assume that $5+3\sqrt{2}$ is a rational number, which can be represented in $\frac{p}{q}$ form and where $q\ne 0$

$$\begin{gathered} \Rightarrow 5+3\sqrt{2}=\frac{p}{q} \\ \Rightarrow 3\sqrt{2}=\frac{p}{q}-5 \\ \Rightarrow \sqrt{2}=\frac{p-5q}{3q} \end{gathered}$$

$\therefore \frac{p-5q}{3q}$ is a rational number

However, we know that $\sqrt{2}$ is an irrational number

This leads to the contradiction that $5+3\sqrt{2}$ is an irrational number

Hence, the given number is an irrational number.